Using the Escalator Boxcar Train to Determitl

the Optimal Management of a Size-Distributer '

Forest when Carbon Sequestration Is Taken

into Account

Renarr Goetzl. Natali Hritonenko2, Angels Xabadial' and Yuri Yatselrk'

I Department of Economics' University of Girona'

Campus clc Nlonti l ivi ' Girona 17071' Spain

renan. goetz@udg ' edu2 Department of \"Iathematics' Prairie \rierv A&\'I Universrty'

Box 519' Prairie Vierv' TX 774'16' USA

t College of Business ancl Economics' Houston Baptist Universitv'

75t)2 Fondren Road' Houston' TX 77074' USA

Abstract. We present a theoreticai model to determlnc the optinral

managementofasize-distr ibrr tedforest .Thedecis ionnrode} isgi l 'eninform of a distributed optirnal control problem that cannot be solved an-

alyt ical ly.Thus,thepapelpresentsarrrrrrrer ical techniquethatal ]or 'n 'stransforming tt'" t"igitttot distributecl control problem into arr ordirrarv

control problem. The"methocl has the aclvantage that it does not require

progralnmlng numerical algorithms btt ratlier can be implernented u'ith

standard .o-ltt"'"iut optiJization packages such as GA\'{S' The empir-

ical application of our model for the case of forest management allolvs

deterrnining the selective cutting regime lr'hen carborr sequestration is

taken into account'

1 Introduction

The anal)'sis of the optirnal management of reneu''able resor"rrces has a ven- I'

historv r,vithin ".or.otrli"''

Traclitionaily' the rener'r'able resource is consicler

asanon-structrrredpopulat iorr . i .e.al l i r ic l i r , idualsareident ical .However. :economic literature t u, ,,ro.u recognized that the assumption of non-structul''

populat ionsdoesnotal lo l l , toaddresscorrect l .vthetr , r .okeyissrtesofthenl i : ,agementofrenervablenatr i ra l resoulces. thedeterminat ionoftheopt imal 'placement periods, ancl the optirnal long-l'un trllocation of the total popr-rlat'

an]ongthedi f t 'erentvaluesoft l restruct t r r ingvar iable(Kenned.-v '1987). In: .case of fbrestr-v, economic a'na]1'sis basecl on non strlrctured populations carii-

correct lydeterninetheopt inralreplacernentper iocls incei tdoesrtot take:cou.t of the f.act that the birth, growth and death processes of the poprilat

clepend on the stmcttlring characteristic-of each individual'

\ ' {oreor,er, thedistr ibut ionoftheincl i r , ic lualcharacter ist icsovertheer i tpopulat ionis i rnport 'ant tomodelcorrecth ' . t l r< ' rnul t ip lcselv icesthatforeststrr

LLirkor ' ,Sl largenor ' 'and' lWaSnieu'ski( t rds):LSSC2007'LNCS'1818'pp3i343'11'20f l8

@ipti.g"r-\ '"rlag Berlin Heidelbcrg 2008

The Escalator Boxcar Train and Forest Manasement JJi)

-:r:

amenity and recreational values, natural habitat of wildlife, mushrooms,:,rtection of watersheds, carbon sequestration etc. A simple example is the-:-rduction and timber and sequestration of carbon. The maximization of the:-., benefits with respect to timber calls for growing a reduced number of high'-:.lue trees while maximization of the net benefi.ts with respect to carbon requires:raximizing the standing biomass of the forest. Thus, a specific distribution of the'',dividual characteristics over the entire forest is necessary to favor a particular,,rest service.

In this paper we present a theoreticai model to determine the optimal seiec----,-e-logging regime of a size-distributed forest. The law of motion of the economic- rdel is governed by a partial integrodifferential equation that describes the

--,',rlution of the forest stock over time. Given the complexity of the problem it is

:,,t possibie to obtain an analytical solution. To solve the problem numericallyi': employ a technique known as the "Escalator Boxcar Train". This technique:.--. been utilized so far to solve partiai differential or partial integrodifferential:,j;-lations describing the evolution of biological populations. In this paper we:-ovide the necessary changes to utilize this method within the context of a,-=tributed optimal control problem. The usefulness of the "Escalator Boxcar-:ain" is illustrated by determining the optimal seiective-logging regime of a: ,r'ately owned forest when carbon sequestration is taken into account.

The paper is organized as follows. The foltowing section presents the economic--rision problem. Section 3 presents the numerical technique of the "Escalator: ,xcar Train" and derives the necessary changes that allow to employ it for the

' -ution of the distributed optimal control problem. Section 4 briefly illustrates';-= proposed numerical technique. The paper ends with a summary.

I The Economic Model

-.note the diameter of a tree by I e Q, O : llo,l-), where 16 and l^ indicate'-,= biological minimum and maximum size of a tree. The decision problem of-"= forest owner, given a planning horizon of f1, can be stated as:

f r t f t ' " f t r

1nax I I B(u(t . t ) . r ( t . t )1erp-, t dt dt - | Ctpu.ts l )erp ,1 dt qt |u(t . t \ .pr1.1o1 1o Jtn ' J0

rI''+ I ,5/ ,1r1t1 . l )yerp-, t ' d l

Jlo

"-rject to the constraints

) r ( t , l ) _at

0(s(e(t) ,1) r ( t , l ) )- 5(e(t) ,1) r ( t , l ) - u( t , l ) , (2)

r ( ts, l ) : ,o( l ) , g(e(t) , to) r ( t , ts) : p( t , tg) , u( t , l ) > 0, p( t , to) > 0, (3)

r:-ere r(f,l) is the state variable that characterizes the distribution density of:.es, and u(t,I) p(t) are the control variables that denote the flux of logged:.es. and the flux of trees planted with diameter l0 respectively, g(e(t),1) is the

AT

336 R. Goetz et al

grou,'th rate of trees. 6(e(f),1) is the instantaneous n-rortality rate. e(l) cle-_the competition betu''cen individuals in the fbrest, that is rneaslrecl by tlie ,

area, that is , e( t ) : I " : " ( i ) r f t . l ) r1t . a 'd r denotes the c l iscount rate

lirnction B(.u(t, l),r(.t.1))e.rp-'t prescnts the disco*ntecl net benefits of ,rnalragement, including the benefits from timber and sequesterecl carbonstrictly convex function c(p)erp rt , g2 captures the cliscountecl cosr or 1,ing trees. the function st' (r)e,st rtt 6 gr the r,aruc of the standing tr.e r -the end of the plan'ing horizon. The term n0(l) clenotes the init ial distrib -densi ty of t rees, and the rest l ic t ion g(e(t) , t0) r ( t . to) : p( t . / ' ) requires thaflux of the change in diameter at l0 mlrltiplied the tree density coirrcides u,itrtotal amount of trees planted lvith diameter ls. Finallv, it is recluired tha.control variables are nonnegative. Equation (2) provicles the equatiori of nr, -for size-structured populations in thc case of a manageci forest lvith no biolo-reproduction (i.e., all )roung trees are planted).

3 The Numerical Approach

To numerically solr,'c the distributed optimal controi problem \ve propo::method rramed the Escalator Boxcar Tlain (de Roos, 198g). In contrast to ot:alailablc methods (Feichtinger, Prskarvetz and \,'elior'. 2004), the Escalator Bcar Train, EBT, can bc implernented r.vith starrdard computer software utili;for solving mathematical programmirrg problerns.

Thc partial intcgrodiffcrential equation (2) descrilres the time evolutionthe population dcnsity-function over the clornain O of the individual stnicrL.variable l. For the dcrivation of the EBT rnethod wc rn'ill assume that eltconstant such tha,t the partial integrodifferential equation is a partial clifleler_equation. Hou,ever. as de Roos (1988) specifies, the rnethod is also applical_,-the case rvhcre the environment is not constant over time. Lct o bc par.titliinto n, * 1 subdomains O.t( t : O) : l / r (O),1,+r(0)) , i :0,1,2. . . . . t . r at the i r rpoint of t ime of thc planning horizon and define f/;(l) as Oi(t) : { l(f. -0. l0)l/0 € oi(O)j, t.e., Qift) defines the cvolution of the tree cliameter over r.,such that all trees that form a coholt in a subdomain at t:0 stay tosethroughout thcir 1ife.

Let the total nurnber of trees, xr(l). and the arreragc diameter of the tr.tz ' ; ( l ) rv i th in the subdomarn Qi, ' i : r ,2, . . . ,n, that we cal l i r r ternar cohor. ts.dcfined as

x,(l)

To take account of the control variable, let U, denote the arnonnt or cLrr rrcc:the subdornain .O;. That is,

: [ ,1t.t1at, andJQi

Lt(t) : ;6 l,,t

r(.t.t)d,l.

, , - [ 'uU.t)cI IJ {);.

The Escalator Boxcar Train and Forest N{anagement 337

totai number of treeb and tlreir avelage dianieter totally characterizes the'LLation u'ithin the subdomain or cohort i. The total population is thus a

--ction of cohorts. \{athematicall l ' , the distribution density r(t, l) ls approxi-

"d b;, a set of dclta functions of size X;(f) at diamcter tr;(t). Using Leibnizr he change in t inre of X, ( t ) is g iverr hy:

' t t , l f ' - ' , , , , f ' '0.111 . t1 , , c l l ; * t , .d l ;. ,1, . l

r ( t ' t ) r t t - J, , A

t t t -L( t ' t , ' \ - i

- t (1 ' t ; \ - (0)

dl ; , , l l ;+tT: .qf"

/ ; ) rnd ; :

o(e.1,-1 ) b, def in i t iorr . and r ts inq lhe eqrral i tv

--nted in equation (2), equation (6) lcads to:

ry# [ , ' ,u,e. t ) r r t . t ) t t r t . t ) ) ( l t t7)

. similar fashion u,e obtain thc ordinary differential equation describing the:rnics of tr1 (f). Frorn now on! the argr,rrnents of the functions u'ill be sup--.ed unless it is required fbr an unambiguous notation.

,lLi _dt

1 fL,+,srdt Tl

( t r , ) (a '+u)dtr r t , . t I l

cl l fL-- tdt xi Jt,

, ( f 'x, \ / i ,

1 f l i - t

iJ,,1 f l ;+t

iJ,,1 f lo+t

N J,,

' * ' I *dL

' ,0.r , , dt , , t d/ , \l . ,d l ) l ; - t . I

- - l ; f

- |

ot ot o, /-+1,,''.'tmtt+

t f fat++1,,". ' \)"-*t,Tr( 6t u)at + + 1,,'".' s rru - *t,*

(8)

- rre ordinary differential equations fbr Xi(l) and Zi(f) described in equations.nd (8) do not forrn a solvable system because they involve rveighted intcgralsilre delsit) ' functiol r(t, l. To obtain a closed solvabie system the functions.r

arid g(e, l) are approxirnated by their first order Taylor expansion around

- I ,(l). Higher order terms involving squales and higher po\vers of (l - ,L6) are

- ' tcd. In this rval-, equation (7) can bc approximated to:

dX ( t \

. - b(e.L; \X;( t ) U;( t \

-ncl equation (8) is approximated to:

(e)

d,L.i(t)

d,t - s, L, - + ( t tudt - t , r ) (10)

R. Goetz et al.

Define t/z as the average diameter of logged trees in cohort i, i.e., Ci : LVl

r I t+t

I t"tt, l)dl, and substitute it into equation (10). Assuming that the cutI .

iJ6", fru,r" in average the same diameter than the standing trees in the cohort.

that is, Ct: Lt, equation (10) leads to

4!9: s@,1)at

(11

Equations (9) and (11) describe the dynamics ofinternal cohorts but they do not

account for the plantation of new trees. The boundary cohort is characterized

byf lr ^ f lt

xo(t): I rf t ,s) as and Loft): I t t - lo)r(t, I) dI. (12J t" J Io

Differentiation of Xs (t) ana i61t; with respect to time, and employing first order

Taylor approximations of the functions g(e, l) and 6(e, l) leads to a set of ordinar-r-

differential equations that describe the dynamics of the boundary cohort' Ther

are given by

d\o =-d(e, ro)xo(t1 - l i ,p, to)Lor;) + p(t)dt" \"- 'Ul-_U\ ' /dL'

dLn r@,to)xo\t) + 4g@,to)L, f t ) - 5(e, Io)Lo(t) . (13

dt =s'

dI- '

In this wav, the partial differential equation and its boundary condition (2

governing the dynamics of the distribution density can be decoupled to a systen'

of ordinary differential equations in xi(t) and Li(t) described by equations (9 "(11), and (13).

Once the partial integrodifferential equation describing the evolution of tbe

system has been transformed into a system of ordinary differential equations. the

objective function needs to be transformed also. Thus, the approximate decisio-

problem is given by:

r ls f l t

, f rH,, / , s(Xtt l .Lft) ,u(t))"rp 'L dt-

Jo "rrr ,))erp ' ' dt (D-

+ ,s/ '1X1t ' ) ,L( l1))erp ' t '

subject to the constraints

4:! : -b(E(r) . L i )x i ( I ) - u i t t ) . 4!9 : s(E(t) . L i )dt

-1 t^^L\- / - . \ - / - dt

+ :-b(E(r) . 10)x0(t) - *or ' (L) . tdLo + p(t)

# : s@ft) .10)x0(,) + f is te{ i lJ0)r0(r)- 6(E(r) .10)ro(t)

Xi@) : X0, s@(t),10)xg(t) : p(t) , Ui( t) ,p(t) > 0 Ui( t) < Xi( t)-

The Escalator Boxcar Train and Forest Management 33g

,, ' : ,X, f__i) and e denote the vectors X : (Xo, . . . , Xn), L : (Lo, . . . , L,)

'- Lt : (Ut,.-.,Un) respectively, X0 the init ial density of each cohort, and- is the approximation of e(f), that is, E(t) : Di "(*)t

,n.

-: is only left one additional transformation. Since lhe'value of the upper.:r al bound of the borrndary cohort, 11, increases over time, arl newly planted=. have a length widening interval llo, lr]. Thus, the boundary cohort cannot be::nued indefinitely, because the range would become larger and larger and theuuuurlre lzlr'ger arlu IaIger anO tne:'oximation would break down. Therefore, the cohorts have to be renumbered=gular time intervals Af . This renumbering operation transforms the current-:rdary cohort into an internal cohort and initializes a new, empty boundary

.- -,r't realizing the following operations

, t^t+) : Xrt(kAt-) , Li(k\ t+) : L. t(kAt-) , Xr(k\r+) : xo(trnr-); , , "{A/ ) : /o*t ' l fa" i - ) t xo(kLt-) =0. Lokat-) :0.k - t .2. . . . (14)" XIGL, )

,: this transformation has been appiied, the system of differential equations' be converted into a system of difference equations that can be solved using-:1ard software such as GAMS (General Algebraic N{odelling system, (Brooke.

-,lr ick and N{eeraus, 1992)).

Empirical Study

: purpose of the empirical anaiysis is to illustrate the applicability of theT method by determining the optimal selective-logging regime of a diameter-:'ibuted forest, that is, the selective logging regime that maximizes the dis-:ted private net benefits from timber production and carbon sequestrationstand of pinus sylueslrzs (scots pine), over time horizon of 300 years.

- ire initial diameter distribution of the forest is presented in Table 1 and.rneters and functions used are defined in Table 2. The thinning and planting,d. Af is set equal to 10 years, which is a common practice for a pi,nus.slrds fbrest.

-:re optimization problem was programmed in GAMS, and the conopt2 solver' employed to find the numerical solution. optimizations with different ran-

initializations of the control variables were carried out to assure that solu-. are independent from the initially chosen values for the numerical optimiza-

.- iechnique. The empirical analysis calculates the optimal selective-logging--:',e. given the initial diameter distribution of the trees described in Table 1- '- discount rate of 2%o. Tabre 3 summarizes the results for the case where

,:r sequestration is not considered and for the case where it is considered.

Table L. Initial diameter distribution of the trees used in the empirical analysis

umber of trees 287 2r8Average diameter (cm) 7.5 12.5

167 11517.5 22.5

68 35 1627.5 32.5 37.5

731142.5 47.5 52.5 57.5

340 R. Goetz et al.

Table 2. Parameters and functions used in the empirical analysis

growth function e(E. l ) : (80 l ) (0.0068983 - 0.00003107 E)6(E, l ) : 9.61rnortality rate:

marketablevolume (m3): (0.699+0.000,111r)(0.0002g4gL2.167563)

t imber price (euros/m3): minl-23.24, + 13.63rr1l '?, g6.651carbon price (euros/Ton of

sequestered CO2): 15logging costs 23.4 euros/m' + 3.6 euros/harnaintenance costs (euros/ha)": 0.07X + 1.18 10-rX2planting costs (euros/trec): 0.60" Nlaintenanee cosls . l "p"nd on l he lotal

Table 3. optimal selective-logging regime (where a discount rate of 2% is assume.:

Optimal management for tirnberYear Number Net tirnber Discounted

oftreesl") revenue(ol net bencfi t(truros/ha) (Euros/ha)

o xs@

Optimal management for timber and c.ar-],Nl. \umber Net l i r r rber (-arbon Discour- iof trees revenue revenue net bene

(Euros/ha) (Euros/ha) (Euros/h.476.09 -44.:J7 -391.;,,

10 94820 97730 99340 997

587.71 - t75.2,1615.18 -135.54892.85 33.961128.42 259.82

50 965 2543.29 628.8560 936 2629.98 547.0570 845 3318.32 638.49

760.82 205.971296.09 193.431938.56 137.952494.13 45.091856.15 -69.68374,4.92 12.2:J

t53.62 -262.137.2r

305..=511. E :596.1255.t,728 3_

9061056TI2811551 1551 1281 1381040

402.72

300 926 2018.79 3.18 1120 2241.93 -77.19 3.1_(") The number of trees irr the forest is calculated just after the trec-

planted, and before the thinning takes place.(b) A11 rnonetary'alues apart from the discounted net benefit are expres:.j

current values.

It shorvs that the incorporation of carbon sequestration benefits i1 the clei.problem leads to an increase in the optimal numbcr of trees in the forest.consequence, the average diameter ofthese trees is lorvcrl . Icadins- to a detrin reverrues f lom t i rnber sale. Howev, ' r . r l re leverrucs f lom the seqrrcster i r rgbon compensates the smallcr t imber revenlles ancl the total surn of disco..nct benefi ts (NPV) of t l ie forest over 300 years is 2gT2.02 euros per ire , -which is a 10% higher than the NPV obtained u,'herr the carbon sequestr-ar-not accounted fbr.

5 Summary

This paper presents a theoretical model that allou's determining the tmanagement of a diametcr-distributed forest where thc gror,r'th proc€-s:'=--' Due to space limitations, the average diameter is not shor,r,n in Table 2.

The Escalator Boxcar Traln and Forest Manaqement 34r

ft:-s deDends not onlv on individual characteristics but also on the distribution: .he individual characteristics over the entire population. The corresponding:,,:omic decision oroblem can be formulated as a distributed ootimal control

pr,blem where the resulting necessary conditions include a system of partialfurr.grodifferential equation that cannot be solved analytically. For this reason)rmiE utilization of a numerical method (Escalator Boxcar Train) is proposed. TheEs:alator Boxcar Train method allows to transform the partiai integrodifferential

:-lation into a set of ordinary differential equations and thereby to approximaterr,ie distributed optimal control problem by a standard optimal control problem.Ih" :ontrast to the existing literature, the resulting optimization problem canhc ::lved numerically utilizing standard mathematical programming techniques*. i does not require the programming of complex numerical algorithms.

-tIr empirical analysis is conducted to show how the EBT method can befmliemented, determining the optimal selective-logging regime of a diameter-,d-;=.:ibuted forest when carbon seouestration is taken into account.

References

Brooke, A., Kendrick, D., Meeraus, A.: GAMS: A User's Guide, release 2.25. Theicientific Press, San Francisco (1992)ie Roos, A.: Numerical methods for structured population models: The EscalatorBoxcar Train. Numer Methods Partial Diff. Equations 4, 173 195 (1988)Feichtinger, G., Prskawetz, A., Veliov, V.: Age-structured optimal control in popu-

-ation economics. Theor. Pop. Biol. 65, 373-387 (2004)

Kennedy, J.: Uncertainty, irreversibility and the loss of agricultural land: a recon-. ideration. J. Agric. Econ. 38, 75 81 (1987)